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Abstract Because of their buoyancy, rigidity, and finite size, inertial particles do not obey the same dynamics as fluid parcels. The motion of small spherical particles in a fluid flow is described by the Maxey–Riley equations and depends nonlinearly on the velocity of the fluid in which the particles are immersed. Fluid velocities in the ocean often have a strong small-scale turbulent component which is difficult to observe or model, presenting a challenge to predicting the evolution of distributions of inertial particles in the ocean. To overcome this challenge, we assume that the turbulent velocity imposes a random force on particles and consider a stochastic analog of the Maxey–Riley equations. By performing a perturbation analysis of the stochastic Maxey–Riley equations, we obtain a simple and accurate partial differential equation for the spatial distribution of particles. The equation is of the advection–diffusion type and handles the uncertainty introduced by unresolved turbulent flow features. In several numerical test cases, distributions of particles obtained by solving the newly derived equation compare favorably with distributions obtained from Monte Carlo simulations of individual particle trajectories and with theoretical predictions. The advection–diffusion form of our newly derived equation is amenable to inclusion within many existing ocean circulation models. Significance StatementWe introduce a new model for describing spatial distributions of small rigid objects, such as plastic debris, in the ocean. The model takes into account the effects of finite particle size and particle buoyancy, which cause particle trajectories to differ from fluid parcel trajectories. Our model also represents small-scale turbulence stochastically. 

Abstract The Indo-Pacific Ocean appears exponentially stratified between 1- and 3-km depth with a decay scale on the order of 1 km. In his celebrated paper “Abyssal recipes,” W. Munk proposed a theoretical explanation of these observations by suggesting a pointwise buoyancy balance between the upwelling of cold water and the downward diffusion of heat. Assuming a constant upwelling velocity w and turbulent diffusivity κ , the model yields an exponential stratification whose decay scale is consistent with observations if κ ∼ 10 −4 m 2 s −1 . Over time, much effort has been made to reconcile Munk’s ideas with evidence of vertical variability in κ , but comparably little emphasis has been placed on the even stronger evidence that w decays toward the surface. In particular, the basin-averaged w nearly vanishes at 1-km depth in the Indo-Pacific. In light of this evidence, we consider a variable-coefficient, basin-averaged analog of Munk’s budget, which we verify against a hierarchy of numerical models ranging from an idealized basin-and-channel configuration to a coarse global ocean simulation. Study of the budget reveals that the decay of basin-averaged w requires a concurrent decay in basin-averaged κ to produce an exponential-like stratification. As such, the frequently cited value of 10 −4 m 2 s −1 is representative only of the bottom of the middepths, whereas κ must be much smaller above. The decay of mixing in the vertical is as important to the stratification as its magnitude . Significance Statement Using a combination of theory and numerical simulations, it is argued that the observed magnitude and shape of the global ocean stratification and overturning circulation appear to demand that turbulent mixing increases quasi-exponentially toward the ocean bottom. Climate models must therefore prescribe such a vertical profile of turbulent mixing in order to properly represent the heat and carbon uptake accomplished by the global overturning circulation on centennial and longer time scales.